{"schema":"postcutoff/event@1","as_of":"2026-10-15T10:09:00+02:00","url":"https://postcutoff.com/e/2026-03-30-erdos-650-exact-value-chatgpt-aristotle/","md":"https://postcutoff.com/e/2026-03-30-erdos-650-exact-value-chatgpt-aristotle/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-03-30-erdos-650-exact-value-chatgpt-aristotle","date":"2026-03-30","date_precision":"day","short_title":"Erdős Problem #650 solved exactly, f(m) = min(m, ⌈2√m⌉), with a strategy proposed by ChatGPT and a Lean proof by Aristotle; first Justin Sun Prize awards in October","deck":null,"takeaway":"On 30 March 2026 Wouter van Doorn, Yanyang Li and Quanyu Tang posted arXiv 2603.28636, which determines exactly the function in Erdős Problem #650 (matching integers to distinct multiples in an interval): f(m) = min(m, ⌈2√m⌉).","category":"science","category_label":"Science & math","importance":2,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2603.28636: Optimal bounds for an Erdős problem on matching integers to distinct multiples","url":"https://arxiv.org/abs/2603.28636","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Erdős Problem #650 (erdosproblems.com)","url":"https://www.erdosproblems.com/650","type":"docs","group":"primary","domain":"erdosproblems.com"},{"n":3,"title":"Formal Conjectures: ErdosProblems/650.lean","url":"https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/650.lean","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"Justin Sun Prize","url":"https://hejustinsun.com/prize","type":"official","group":"primary","domain":"hejustinsun.com"},{"n":5,"title":"The Block (sponsored): Justin Sun Prize's New Round Honors Human–AI Collaboration on Erdős Problems","url":"https://www.theblock.co/sponsored/2026-10-05-justin-sun-prizes-new-round-honors-human-ai-collaboration-on-erdos-problems-417743","type":"press","group":"press","domain":"theblock.co"}],"official":4,"filed":"2026-10-14","updated":"2026-10-14","orgs":["OpenAI","Harmonic"],"title":"Erdős Problem #650 solved exactly, f(m) = min(m, ⌈2√m⌉), with a strategy proposed by ChatGPT and a Lean proof by Aristotle; first Justin Sun Prize awards in October","summary":"On 30 March 2026 Wouter van Doorn, Yanyang Li and Quanyu Tang posted arXiv 2603.28636, which determines exactly the function in Erdős Problem #650 (matching integers to distinct multiples in an interval): f(m) = min(m, ⌈2√m⌉). The abstract says the proof strategy \"was first proposed by ChatGPT\" and the argument was \"made fully rigorous and formally verified in Lean by Aristotle\"; the exposition is human-written. On 5 Oct 2026 the three were named the first recipients of the crypto-funded Justin Sun Prize for this and related Erdős work (sponsored press release; no amount stated).","key_facts":["Paper: 'Optimal bounds for an Erdős problem on matching integers to distinct multiples', arXiv 2603.28636 (math.CO, math.NT), submitted 30 March 2026, 8 pages; Wouter van Doorn, Yanyang Li, Quanyu Tang","Problem (paper): f(m) is the largest integer such that for every set A of m positive integers and every open interval I of length 2a_m there are at least f(m) disjoint pairs (a, b) with a ∈ A dividing b ∈ I. Result: f(m) = min(m, ⌈2√m⌉) for all m","History (erdosproblems.com/650): Erdős and Surányi (1959) proved f(m) ≥ √m; Erdős and Selfridge gave f(m) ≤ 2⌈√m⌉; GPT 5.4 Pro, prompted by He, Li and Tang, proved f(m) ≤ ⌈2√m⌉, and GPT 5.4 Pro with Aristotle gave a matching lower bound; the site lists the problem as solved by [VLT26]","AI disclosure (abstract): 'The proof was obtained through an AI-assisted workflow: the proof strategy was first proposed by ChatGPT, and the detailed argument was subsequently made fully rigorous and formally verified in Lean by Aristotle. The exposition and final proofs presented here are entirely human-written.'","The problem's formal statement is in Google DeepMind's Formal Conjectures repository (ErdosProblems/650.lean)","Justin Sun Prize (announced Geneva, Oct 5, 2026; carried as TRON-commissioned sponsored content by The Block and Cointelegraph): van Doorn, Tang and Li are the program's 'first confirmed awards', for six Erdős problems: #650; #369, #457 and #469 (van Doorn, Lean-checkable proofs); #1044 (Tang); #1196 (Tang and Li within a wider team). Paid in USDT on TRON or USDC on Ethereum; no amounts or judges named","Prize site (hejustinsun.com/prize): motto 'Quod probatur, solvitur'; accepts complete Lean proofs, solutions without Lean, or previously catalogued work; no distinction between human and AI contributors"],"key_numbers":[],"tags":["math","number-theory","combinatorics","erdos-problems","chatgpt","aristotle","lean","formal-verification","ai-assisted","prizes"],"science":{"field":"mathematics","subfield":"combinatorial number theory","problem":"Erdős Problem #650: how many integers of a set A can always be matched to distinct multiples inside an interval of length 2·max(A)","result":"f(m) = min(m, ⌈2√m⌉) for all m, settling the problem exactly","open_since":"1959","ai_system":["ChatGPT","GPT-5.4 Pro","Aristotle"],"human_role":"AI-assisted: ChatGPT proposed the strategy, Aristotle produced the Lean-verified argument; the authors simplified it and wrote the paper","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean (by Aristotle, per the authors); arXiv preprint"},"status":"confirmed","shock":"","short":"Erdős #650 exact value"},"body_md":"## What happened\n\nErdős Problem #650 asks, for a set of m integers, how many of them can always be paired with distinct multiples lying in a short\ninterval. Erdős and Surányi had shown at least √m in 1959. In early 2026 GPT 5.4 Pro, prompted by researchers, improved the upper\nbound to ⌈2√m⌉ and, with Harmonic's Aristotle, found a matching lower bound (as recorded on erdosproblems.com). On 30 March 2026\nWouter van Doorn, Yanyang Li and Quanyu Tang posted the exact answer, f(m) = min(m, ⌈2√m⌉), stating in the abstract that ChatGPT\nproposed the proof strategy and Aristotle made it rigorous and verified it in Lean, while the final text is human-written.\n\nThe result returned to the news in October 2026, when the Office of Justin Sun announced in Geneva (Oct 5) that the three were the\nfirst recipients of the Justin Sun Prize, a crypto-paid prize for Erdős problems that accepts Lean proofs and does not distinguish\nhuman from AI contributors. The announcement circulated as TRON-commissioned sponsored content (The Block, Cointelegraph) and names\nno amounts or judges.\n\n## Why it matters\n\nA small, clean example of the 2026 workflow for open problems: a chat model proposes the idea, a formal prover closes the gaps in\nLean, and humans simplify and write it up. The prize shows new money (here from a crypto entrepreneur) attaching itself to\nAI-assisted Erdős results, which the erdosproblems.com maintainers had begun to ration by October (see the related entry on the site\nfreezing proof claims).","disputed":[],"related":[{"id":"2026-10-06-erdos-problems-site-freezes-proof-claims","url":"https://postcutoff.com/e/2026-10-06-erdos-problems-site-freezes-proof-claims/","date":"2026-10-06","date_precision":"day","short_title":"erdosproblems.com freezes proof claims and drops 'open/solved' labels and solver credits after a wave of unexplained AI proofs","deck":null,"takeaway":"erdosproblems.com was where AI-for-math claims were counted and disputed, from the GPT-5 controversy of October 2025 to the 2026 waves of GPT-6 Astra and Claude results.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":4,"official":4,"filed":"2026-10-08","updated":"2026-10-12","orgs":["erdosproblems.com"]},{"id":"2026-05-03-erdos-1196-primitive-sets","url":"https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/","date":"2026-05-03","date_precision":"day","short_title":"Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets","deck":"Tao co-authors the paper","takeaway":"It was the first AI solution to a well-known, decades-old Erdős conjecture that specialists had actively worked on, not just an obscure entry.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]},{"id":"2025-12-08-ai-erdos-problems-wave-late-2025","url":"https://postcutoff.com/e/2025-12-08-ai-erdos-problems-wave-late-2025/","date":"2025-12-08","date_precision":"day","short_title":"Genuine AI-assisted solutions to Erdős problems begin","deck":"#124 (Aristotle), #1026 (48-hour human–AI collaboration)","takeaway":"Erdős problems became the first large, public, verifiable scoreboard for AI in research mathematics, and set the stage for 2026's much larger results.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Harmonic","Google DeepMind","OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-14","type":"filed","text":"Created (arXiv abstract, erdosproblems.com page and the sponsored release were read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-14","run":null,"sources_read":"ArXiv abstract, erdosproblems.com page and the sponsored release were read","updated":"2026-10-14","human_review":null,"version":{"date":"2026-10-14"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":null,"in_training_data":true},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":null,"in_training_data":true},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":null,"in_training_data":true},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":null,"in_training_data":true}],"short_url":null}